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Zipf's law states that given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table. Thus the most frequent word will occur approximately twice as often as the second most frequent word, three times as often as the third most frequent word, etc. For example, in the Brown Corpus of American English text, the word "the" is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences (69,971 out of slightly over 1 million). True to Zipf's Law, the second-place word "of" accounts for slightly over 3.5% of words (36,411 occurrences), followed by "and" (28,852). Only 135 vocabulary items are needed to account for half the Brown Corpus.[4]

The same relationship occurs in many other rankings unrelated to language, such as the population ranks of cities in various countries, corporation sizes, income rankings, ranks of number of people watching the same TV channel,[5] and so on. The appearance of the distribution in rankings of cities by population was first noticed by Felix Auerbach in 1913.[3] Empirically, a data set can be tested to see whether Zipf's law applies by checking the goodness of fit of an empirical distribution to the hypothesized power law distribution with a Kolmogorov-Smirnov test, and then comparing the (log) likelihood ratio of the power law distribution to alternative distributions like an exponential distribution or lognormal distribution.[6] When Zipf's law is checked for cities, a better fit has been found with b = 1.07; i.e. the largest settlement is the size of the largest settlement. While Zipf's law holds for the upper tail of the distribution, the entire distribution of cities is log-normal and follows Gibrat's law.[7] Both laws are consistent because a log-normal tail can typically not be distinguished from a Pareto (Zipf) tail.

Zipf's law is most easily observed by plotting the data on a log-log graph, with the axes being log (rank order) and log (frequency). For example, the word "the" (as described above) would appear at x = log(1), y = log(69971). It is also possible to plot reciprocal rank against frequency or reciprocal frequency or interword interval against rank.[1] The data conform to Zipf's law to the extent that the plot is linear.

Formally, let:

N be the number of elements;

k be their rank;

s be the value of the exponent characterizing the distribution.

Zipf's law then predicts that out of a population of N elements, the frequency of elements of rank k, f(k;s,N), is:

Zipf's law holds if the number of occurrences of each element are independent and identically distributed random variables with power law distribution [8]

In the example of the frequency of words in the English language, N is the number of words in the English language and, if we use the classic version of Zipf's law, the exponent s is 1. f(k; s,N) will then be the fraction of the time the kth most common word occurs.

The simplest case of Zipf's law is a "1⁄f function". Given a set of Zipfian distributed frequencies, sorted from most common to least common, the second most common frequency will occur ½ as often as the first. The third most common frequency will occur ⅓ as often as the first. The nth most common frequency will occur 1⁄n as often as the first. However, this cannot hold exactly, because items must occur an integer number of times; there cannot be 2.5 occurrences of a word. Nevertheless, over fairly wide ranges, and to a fairly good approximation, many natural phenomena obey Zipf's law.

Mathematically, the sum of all relative frequencies in a Zipf distribution is equal to the harmonic series, and

In human languages, word frequencies have a very heavy-tailed distribution, and can therefore be modeled reasonably well by a Zipf distribution with an s close to 1.

As long as the exponent s exceeds 1, it is possible for such a law to hold with infinitely many words, since if s > 1 then

It is not known why Zipf's law holds for most languages.[9] However, it may be partially explained by the statistical analysis of randomly generated texts. Wentian Li has shown that in a document in which each character has been chosen randomly from a uniform distribution of all letters (plus a space character), the "words" follow the general trend of Zipf's law (appearing approximately linear on log-log plot).[10]Vitold Belevitch in a paper, On the Statistical Laws of Linguistic Distribution offered a mathematical derivation. He took a large class of well-behaved statistical distributions (not only the normal distribution) and expressed them in terms of rank. He then expanded each expression into a Taylor series. In every case Belevitch obtained the remarkable result that a first-order truncation of the series resulted in Zipf's law. Further, a second-order truncation of the Taylor series resulted in Mandelbrot's law.[11][12]

The principle of least effort is another possible explanation: Zipf himself proposed that neither speakers nor hearers using a given language want to work any harder than necessary to reach understanding, and the process that results in approximately equal distribution of effort leads to the observed Zipf distribution.[13][14]

A plot of word frequency in Wikipedia (November 27, 2006). The plot is in log-log coordinates. x is rank of a word in the frequency table; y is the total number of the word’s occurrences. Most popular words are "the", "of" and "and", as expected. Zipf's law corresponds to the middle linear portion of the curve, roughly following the green (1/x) line, while the early part is closer to the magenta (1/"x^0.5") line while the later part is closer to the cyan (1/"(k+x)^2.0") line. These lines correspond to three distinct parameterizations of the Zipf-Mandelbrot distribution.

Zipf's law in fact refers more generally to frequency distributions of "rank data," in which the relative frequency of the nth-ranked item is given by the Zeta distribution, 1/(nsζ(s)), where the parameter s > 1 indexes the members of this family of probability distributions. Indeed, Zipf's law is sometimes synonymous with "zeta distribution," since probability distributions are sometimes called "laws". This distribution is sometimes called the Zipfian or Yule distribution.

The "constant" is the reciprocal of the Hurwitz zeta function evaluated at s. In practice, as easily observable in distribution plots for large corpora, the observed distribution can better be modelled as a sum of separate distributions for different subsets or subtypes of words that follow different parameterizations of the Zipf-Mandelbrot distribution, in particular the closed class of functional words exhibit "s" lower than 1, while open-ended vocabulary growth with document size and corpus size require "s" greater than 1 for convergence of the Generalized Harmonic Series.[1]

In the parabolic fractal distribution, the logarithm of the frequency is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship.[15] Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts.[17]

It has been argued that Benford's law is a special bounded case of Zipf's law,[15] with the connection between these two laws being explained by their both originating from scale invariant functional relations from statistical physics and critical phenomena.[18] The ratios of probabilities in Benford's law are not constant. The leading digits of data satisfying Zipf's law with s = 1 satisfies Benford's law.

Benford's law:

1

0.30103000

2

0.17609126

-0.7735840

3

0.12493874

-0.8463832

4

0.09691001

-0.8830605

5

0.07918125

-0.9054412

6

0.06694679

-0.9205788

7

0.05799195

-0.9315169

8

0.05115252

-0.9397966

9

0.04575749

-0.9462848

Zipf's distribution is also applied to estimate the emergent value of networked systems and also service-oriented environments[clarification needed].